Why is the world so screwed up? Is it God's fault?

This is a complicated question. There are lots of layers both in understanding what the Bible says God's perspective is on this subject, and then also how God's perspective impacts the people who are experiencing the brokenness of the world. Here's one effort to offer a simplified explanation of how to think about this:

1. God is love.
2. God made the world and filled it with people that He wanted to love to have love Him back.
3. Love requires free will. If you can't choose, or you are forced to choose, it's not love.
4. Human beings have used their free will to do evil:  to themselves, to other people, and to the world.
5. The good people and the good world have been vandalized and corrupted by the evil things people have done.
6. The evil things people have done has taken control of the world and become almost a ruling power in and of itself, being further propagated in the world. 
7. God loves and cares so much that He has consistently acted to re-engage with the world that has banished Him, to rescue people from the power of evil, and to renew them so that they can fix what has been vandalized and corrupted.
8. God loves and cares so much that He actually entered into the world in Jesus to be a human strong enough to overcome the power of evil in the world. 
9. God and cares so much that He entered into our broken world and experienced the full evil of the world and of people in His suffering on the cross. His suffering was Him taking on the consequences of the evil of people. He did this to bring forgiveness from God for the evil the we do, and to bring power into the world so that other people could receive that power and overcome it in their own lives. His resurrection shows that He is stronger than the world's evil. When we believe and follow Jesus, we receive His power. Overcoming evil in our lives means having the strength to endure the evil and brokenness of the world in our own personal lives and to be able to respond to it all with grace, understanding, love, and the strength to show others a better way.

Christmas is a chance to celebrate that we have a God who didn't destroy evil people from up in heaven, but He came into the world to rescue and renew it.